Such a method and such a user system is known from EP 1 972 959 A1. According to the known method the carrier signals of a global navigation satellite system are processed using linear combinations of the carrier signals for estimating the phase ambiguities and the ionospheric error. Since the ionospheric error is known, the position of the user system can be determined absolutely without using parallel measurements of a reference station.
The known method applies to global navigation satellite system with at least three carriers, such as triple frequency GPS and multi-frequency Galileo measurements. The higher number of frequencies and the optimized modulation of the new Galileo signals and the GPS L5 signal will reduce the noise variance. However, the inherent disadvantages for partial ambiguity resolution are not mitigated. One of them is the current maximization of the reliability of the first fix instead of the maximization of the number of reliably fixable ambiguities. Another drawback is that systematic errors have not been taken into account for the search of the optimal fixing order.
The dispersive behaviour of the ionosphere in the L band can also be estimated within GPS. Currently, the ionospheric delay is estimated from GPS L1 and L2 code measurements which are BPSK modulated. This modulation centers the power around the carrier frequency, i.e. the power spectral density is substantially lower at the band edges than at the band center. This results in a larger Cramer Rao bound than other modulations and, thus, an increased code noise. The increased code noise as well as multipath errors are further amplified by the linear combination for ionospheric delay estimation. A rough estimate is obtained by a simple L1-L2 code-only combination. The weighting coefficients of the dual frequency combination are chosen such that the true range and the non dispersive errors (clock offsets and tropospheric delay) are eliminated and only the ionospheric delay is preserved. The weighting coefficients of a dual frequency code only combination are unambiguously given by the geometry-free and ionosphere preserving constraints. Therefore, there is no degree of freedom to minimize the noise variance. An ionospheric delay estimation with GPS L1 and L2 code measurements therefore requires an ionosphere-free carrier smoothing with large time constants and large filter initialization periods to achieve a centimeter accuracy.
The carrier phase measurements can also be used for positioning and ionospheric delay estimation in addition to the code measurements. The carrier phase measurements are about three orders of magnitude more accurate than code measurements. However, they are ambiguous as the fractional phase of the initial measurement does not provide any information on the integer number of cycles (called integer ambiguity) between the receiver and the satellite.
A ionosphere-free carrier smoothing is a widely used method to reduce the code noise by carrier phase measurements without resolving their ambiguities. A centimeter accuracy for the ionospheric delay estimation can be achieved with time constants and filter initialization periods of several minutes.
An even higher accuracy of the ionospheric delay estimation requires the resolution of the carrier phase integer ambiguities. The reliable resolution of all ambiguities can not be achieved under severe multipath conditions, especially affecting low elevation satellites, such that the fixing is limited to a subset of ambiguities (=partial fixing). There exist various approaches for the estimation of integer ambiguities, e.g. the synchronous rounding of the float solution, a sequential fixing (=bootstrapping), an integer least squares search or integer aperture estimation.
The synchronous rounding of the float solution is the most simple method. However, it does not consider the correlation between the real valued ambiguity estimates which results in a lower success rate and a lower number of reliable fixes than other methods. Moreover, there does not exist an analytical expression for the success rate which has to be determined by extensive Monte Carlo simulations.
The sequential ambiguity fixing (bootstrapping) is another very efficient method that takes the correlation between the float ambiguities into account and, therefore, enables a substantially lower probability of wrong fixing. Another advantage is that the success rate can be computed analytically. However, this success rate and the number of reliably fixable ambiguities depend strongly on the chosen order of fixings. The optimization of the fixing order becomes especially important for geometries with a high number of visible satellites. A drawback of the sequential ambiguity fixing (bootstrapping) is that the success rate is slightly lower than for integer least squares estimation.
The third approach, the integer least square estimation, maximizes the success rate for unbiased measurements and includes an integer decorrelation which enables a very efficient search. A disadvantage of the integer least squares estimation is the lack of an analytical expression for the success rate. It can only be approximated by extensive Monte Carlo simulations. Moreover, the integer least squares estimation is only optimal in the absence of biases.
The success rate of ambiguity fixing depends substantially on residual biases. It is known that these biases degrade the success rate significant although a quantitative analysis has not been made so far. Moreover, the widely used sequential fixing (bootstrapping) uses a fixing order that maximizes the reliability of the first fix (i.e. smallest variance in the float solution). After this first fix, the float solution is updated and the most reliable ambiguity is selected among the remaining ones. This procedure is repeated until a predefined threshold on the probability of wrong fixing is hit or all ambiguities are fixed. The disadvantage of this method is that maximizing the reliability of the first fixes does not maximize the number of reliably fixable ambiguities. Also the other methods, e.g. synchronous rounding, integer least squares estimation or integer aperture estimation, can be implemented efficiently but suffer from a large computational burden for the evaluation of the success rate. It is determined by a large number of Monte Carlo simulations to achieve reliable estimates of the probability of wrong fixing which is in the order of 10−9.